Evolving Algebraic Multigrid Methods Using Grammar-Guided Genetic Programming

TL;DR

This work tackles the limitation of traditional algebraic multigrid (AMG) cycles by introducing grammar-guided genetic programming (G3P) to automatically generate flexible, arbitrary cycling structures. By encoding AMG steps as algebraic expressions and constraining cycle generation with context-free grammars, the authors evolve AMG programs that are then executed within hypre's BoomerAMG framework using the EvoStencils/DEAP optimization pipeline. The evolved solvers and preconditioners consistently outperform standard AMG cycles, achieving faster solve times, improved convergence, and robust performance across varying problem sizes and time-varying matrices. This approach not only yields problem-adapted AMG methods but also provides a dataset and methodology for future ML-driven method selection in AMG.

Abstract

Multigrid methods despite being known to be asymptotically optimal algorithms, depend on the careful selection of their individual components for efficiency. Also, they are mostly restricted to standard cycle types like V-, F-, and W-cycles. We use grammar rules to generate arbitrary-shaped cycles, wherein the smoothers and their relaxation weights are chosen independently at each step within the cycle. We call this a flexible multigrid cycle. These flexible cycles are used in Algebraic Multigrid (AMG) methods with the help of grammar rules and optimized using genetic programming. The flexible AMG methods are implemented in the software library of hypre, and the programs are optimized separately for two cases: a standalone AMG solver for a 3D anisotropic problem and an AMG preconditioner with conjugate gradient for a multiphysics code. We observe that the optimized flexible cycles provide higher efficiency and better performance than the standard cycle types.

Figures (6)

• Figure 1: Visual representation of the cycling structure in a multigrid method. The color of the node denotes the smoother type with its relaxation weights indicated inside.
• Figure 1: Software setup
• Figure 1: Mean and standard deviation of the minimum objective function values during the optimization (left) and combined Pareto distributions (right), over ten experiments.
• Figure 2: Weak scaling of a 3D anisotropic poisson problem.
• Figure 3: Mean and standard deviation of the minimum objective function values during the optimization (left) and combined Pareto distributions (right), over five experiments.